SCHWEITZER 2005 Problem 4

Let F be a countable free group and let $F = H_1> H_2> H_3> \cdots$ be a descending chain of finite index subgroups of group F. Suppose that $\cap H_i$ does not contain any nontrivial normal subgroups of F. Prove that there exist $g_i\in F$ for which the conjugated subgroups $H_i^{g_i}$ also form a chain, and $\cap H_i^{g_i}=\{1\}$ .